The sum of the H.C.F. and L.C.M of two number is 680 and the L.C.M is 84 times the H.C.F. If one of the numbers is 56 , the other is :

Options

A. 84
B. 12
C. 8
D. 96

Answer

To solve the problem step by step, we will use the information given in the question:

1. **Understanding the Problem**: We know that the sum of the H.C.F. (Highest Common Factor) and L.C.M. (Lowest Common Multiple) of two numbers is 680. We also know that the L.C.M. is 84 times the H.C.F. One of the numbers is given as 56, and we need to find the other number.

2. **Setting Up the Equations**:
- Let the H.C.F. be denoted as \( x \).
- Therefore, the L.C.M. can be expressed as \( 84x \) (since L.C.M. is 84 times the H.C.F.).
- According to the problem, we have the equation:
\[ x + 84x = 680 \]

3. **Combining Like Terms**:
- Combine the terms on the left side:
\[ 85x = 680 \]

4. **Solving for H.C.F.**:
- To find \( x \), divide both sides of the equation by 85:
\[ x = \frac{680}{85} \]
- Simplifying this gives:
\[ x = 8 \]

5. **Finding the L.C.M.**:
- Now that we have the H.C.F. (which is \( x = 8 \)), we can find the L.C.M.:
\[ \text{L.C.M.} = 84 \times 8 = 672 \]

6. **Using the Relationship Between H.C.F., L.C.M., and the Numbers**:
- The relationship between the two numbers (let's denote them as \( A \) and \( B \)) is given by:
\[ \text{H.C.F.} \times \text{L.C.M.} = A \times B \]
- Substituting the known values:
\[ 8 \times 672 = 56 \times B \]

7. **Calculating the Other Number**:
- First, calculate \( 8 \times 672 \):
\[ 5376 = 56 \times B \]
- Now, divide both sides by 56 to find \( B \):
\[ B = \frac{5376}{56} \]
- Simplifying this gives:
\[ B = 96 \]

8. **Conclusion**:
- The other number is \( 96 \).